We generalize some fundamental results for noncompact Riemannian manfolds without boundary, that only require completeness and no curvature assumptions, to manifolds with boundary: let M be a smooth Riemannian manifold with boundary \(\partial M\) and let \(\hat{C}^\infty _c(M)\) denote the space of smooth compactly supported cut-off functions with vanishing normal derivative, Neumann cut-offs. We show, among other things, that under completeness: \(\hat{C}^\infty _c(M)\) is dense in \(W^{1,p}(\mathring{M})\) for all \(p\in (1,\infty )\) ; this generalizes a classical result by Aubin (Bull. Sci. Math. 100:149–173, 1976) for \(\partial M=\emptyset \) .
M admits a sequence of first order cut-off functions in \(\hat{C}^\infty _c(M)\) ; for \(\partial M=\emptyset \) this result can be traced back to Gaffney (Ann. Math. (2) 60(1):140–145, 1954).
the Laplace–Beltrami operator with domain of definition \(\hat{C}^\infty _c(M)\) is essentially self-adjoint; this is a generalization of a classical result by Strichartz (J. Funct. Anal. 52(1):48–79, 1983) for \(\partial M=\emptyset \) .
